sobolev embedding theorem in the smooth metric measure space - MathOverflow

sobolev embedding theorem in the smooth metric measure space

2012-04-20T04:49:23Z

we know the sobolev embedding theorem of Saloff-Coste

$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu $

wtih $Ric\ge-(n-1)K$, for all '$B$' of radius $R$ and volume $V$, $F\in C^{\infty}_0(B)$, $q=n/(n-2)$.

My question is whether this inequality was established in the smooth metric measure space,i.e. $(M,g,e^{-f}d\mu)$ with Bakry-Emery Ricci curvature bouneded below $Ric_f=Ric+Hess f\ge-(n-1)K$?

Thank you!

Answer by Robert Haslhofer for sobolev embedding theorem in the smooth metric measure space

2012-04-20T09:44:49Z

The Sobolev-inequality holds for general metric measure spaces satisfying CD(K,n), in particular for your smooth ones.

See e.g. Theorem 21.15 in Villani's book http://math.univ-lyon1.fr/~villani/Cedrif/B07D.StFlour.pdf

Also note that the $L^2$-version of the Sobolev-inequality follows from the $L^1$-version by inserting a suitable power of the function and using Hölder.